Jun 28, 2014   #Mathemusicking  #Children  #Mathematics  #Music  #Cognitive models 

Recent writing in the field of ethnomusicology has re-asked the question of “what is music?”. Christopher Small coined the term “musicking”, which to me expresses that there is no such thing as “music” that is apart from the act of “musicking”. Music and mathematics have shared a historical bond with each other - with mathematicians finding fascination in musical patterns and musicians relishing in artistic construction using mathematical patterns, more recently involving computational patterns. The relationship that both these activities bear to the functioning of human cognition also share great similarities. Mathematicians have long declared the activity of “doing mathematics” as a creative process that is not steeped in certainties, as a naive view of mathematics might suppose. Paralleling that, musicians also often demonstrate intellectualization of the activity of musicking that resembles a mathematical theory of the constructs that they are building. In consideration of such deep connections, in this essay, I explore the parallel thesis - there is no such thing as mathematics, there is only mathematicking - and where I name the joint activity “mathemusicking”.

Top physicists have long pondered the puzzle of why the laws of nature are best expressed in the language of mathematics. This is a question that I believe every scientist faces at some time or the other. Indeed, a younger me has been consumed by this question over weeks, through even sleep, at one point and it has continued to backdrop all my thinking about modern physics and mathematics, though I’m not a professional in either field. The question’s depth feels so inaccessible that we often give it up as a “mystery” soon after we think of that question. At least, I did.

Now, I’ve adopted a point of view from which the question no longer seems mysterious in the way it used to - unfathomable or ineffable - though it has other mysteries to it. This point of view is that the form of mathematics - the “language” as we call it - is inextricably bound to the laws of the universe in which it is effected. Yes, I’d like to claim that a universe with different physical laws would result in correspondingly different mathematics.

If that sounds ridiculous, it may not continue to be so after we step into human cognition for a while and look at what processes might be involved in concept formation. Drescher’s exploration of Piaget’s schema theory of child development in the context of an AI’s development in a simulated world offers an interesting case. He introduces an algorithm called “marginal attribution” for an AI to learn new schemas. In particular, he identifies a “reified schema” as a newly created “item” in the AI’s store that stands for the consequence of a particular schema playing out its action under its context. Having such a “reified schema” available as state is a process by which the system is able to build abstractions about the world it inhabits.

If we ask what having such reified schemas in our cognitive system feels like, I will not be surprised if it feels as through these abstractions have a reality of their own. Indeed, the term “reified” reflects this tendency. The same kind of computational resources are being used to model both “primitive schemas” (those that directly relate to sensory input) and these abstract “reified schemas”. This makes it likely that our cognition would treat both kinds of computational patterns in a similar way. This model explains why Plato and Pythagorus were unable to escape theorizing that pure geometrical objects inhabited their own “planes of existence”.

I therefore suspect that it is entirely possible that those entities that we regard as mathematical certainties, to which we might tend to attribute a reality independent of the physical universe we inhabit, are such “reifications” in our collective cognitive fabric.

This model has several other advantages too. For one thing, such reifications can differ among individuals, especially when in their primitive stages. What we conventionally call mathematics extends this process to its extreme where similar reifications are attempted to be created in all the minds that engage in it. When Hardy and Littlewood looked at Ramanujan’s letters, they found him employing notation and drawing conclusions that they could not comprehend, despite Ramanujan having not learnt any special form of mathematics and working with standard textbooks alone. It took a while for the reifications in Ramanujan’s mind to travel to other minds. While Ramanujan’s offers a striking example, this is a process we all undergo on a daily basis, whether we’re doing mathematics or not.

Musicking has a parallel as well. Though, for most, music is deeply connected with humans as “emotional beings” and a naive view of mathematics as being emotionless does exist in the common mind, some of the most deeply experienced music cannot rightfully be described as “emotional”, in much the same way as a mathematician might describe a “beautiful theory” and, depending on the theory, have an equally powerful experience of ecstasy or epiphany in the act of discovering it. In musicking, we may be said to “communicate” only when we are able to share at least some of these reifications in our cognitive fabric. This definitely places our conventional notion of music as a social construct, and forces us to ask “what if mathematicking is similar too?”.

Our famous works of literature, especially poetry, are rife with the “music” in the speech of children, in the rustling of leaves, in the “rhythmic” bashing of waves on the seashore. “Kuzhal inidu yaazh inidu enbar tammakkal mazhalaichchol kelaadavar” - a Tamil saying that means “Those who say that the flute or the harp is beautiful have not heard their children’s utterances”. What is it that we hear as “music” in the utterances of a child? Our recent discoveries about child development indicate that this same child also explores his/her world in a scientific manner … doing it with such command that they’d put many professional scientists to shame. All you need to do is to watch a child learn how to walk or pick up language, starting from birth. Children differ among themselves in the phases that they go through, based on the affordances of their bodies. However, the way they discover their limits incrementally is fascinating to observe. Could these children be said to be attempting “mathematics to our minds” in the same sense in which we’d call their attempts at speech “music to our ears”?

Whatever the process of “mathematicking” is, there is no escaping the fact that it is something that entities in this physical universe do. However real a mathematical theory might feel to us, we are, at the mercy of reified concepts in our cognitive fabric and the way it feels to have them.

So, how about we marry these two principles – that there is “only musicking” and “only mathematicking” in a simultaneous exploration of both - “mathemusicking”? While the term could be naively interpreted as something like “exploring music through mathematics”, as though mathematics were an instrument like a ship using which we “explore” the oceans of music, how about we think of this as a joint activity, where neither music nor mathematics is merely an instrument that enlightens the other, where both have their own equal standing, where both are grounded in the body and how we use our bodies and minds to sense our world, where, perhaps, we might allow ourselves to be confused about whether we’re musicking or mathematicking, and where we live out our lives in the triangular space between music, mathematics and human cognition?